The Radial Wave Equation

Solutions of the Schrodinger equation can be found which are the product of three functions, each one involving only one of the variables r, 0 and (j . The wavefunction can therefore be written as  

the variables upon which the functions operate have been omitted to reduce the length of the equation. The functions 0(0) and are found to be exactly the same as the wavefunctions which were discussed in Chapter 5 for a particle on the surface of a sphere. These are the spherical harmonics, T (0, 0), which depend upon the two quantum numbers. / and nil. Thus, 0) ( ) = 0). It was shown in Chapter 5 that 

Combining this equation with equation (6.13) and writing Y in place of 0(0) p(0), we obtain  

The radial equation for the hydrogen atom is then obtained by dividing throughout by RY  

Equations similar to equation (6.16) have been studied by mathematicians, and acceptable solutions found. The mathematics involved are quite lengthy and wiU not be given here they can be found in standard textbooks on quantum mechanics. Two quantum numbers are needed to specify a particular radial wavefunction. The first one is the azimuthal quantum number. I, and the second one is a new quantum number , often referred to as the principal quantum number. All the solutions have the same mathematical structure, which can be expressed by the equation  

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Example 5. The diatomic molecule.6 The radial wave equation is... 

This is called the radial wave equation. Apart from the term involving l, it is the same as the one-dimensional time-independent Schrodinger equation, a fact that will be useful in its solution. The last term is referred to as the centrifugal potential, that is, a potential whose first derivative with respect to r gives the centrifugal force. 

The radial wave function has (n — l+l) nodes, where n and l are the quantum numbers. To solve the radial atomic wave equation above, the Herman-Skillman method [12] is usually used. The equation above may be rewritten in a logarithmic coordinate of radius. The radial wave equation is first expressed in terms of low-power polynomials near the origin at the nucleus [13]. With the help of the derived polynomials near the origin, the equation is then numerically solved step by step outward from the origin to satisfy the required node number. At the same time, the radial wave equation is solved numerically from a point far away from the origin, where the radial wave function decays exponentially. The inner and outer solutions are required to be connected smoothly including derivative at a connecting point. 

There are no analytical forms for the radial functions, / ni(r), as solutions of the radial wave equation. Hartree, in 1928, developed the standard solution procedure, the self-consistent field method for the helium atom by using the simple product forms of equation 1.10 to represent the two-electron wave function. Herman and Skillman (4) programmed a very useful approximate form of the Hartree method in the early 1960s for atomic structure calculations on all the atoms in the Periodic Table. An executable version of this program, based on their FORTRAN code, modified to output data for use on a spreadsheet is included with the material on the CDROM as hs.exe. 

The radial wave equation for a particle moving with orbital angular momentum I in a Coulomb field is... 

Understand the origins of the radial wave equation and the significance of the three energy terms that occur within it... 

One procedure for obtaining a solution of the radial wave equation is to make a guess as to the correct form of the wavefunction and see if it works. This is done in the example that follows. 

As one side of the last equation depends only on r, the other one only on 0 and (f), both sides must be equal to a constant, say A. Thus, one arrives at the "radial wave equation ... 

The radial wave is tire solution of the radial Sclirodinger equation... 

Solution of the Schrodinger equation for R i r), known as the radial wave functions since they are functions only of r, follows a well-known mathematical procedure to produce the solutions known as the associated Laguerre functions, of which a few are given in Table 1.2. The radius of the Bohr orbit for n = 1 is given by... 

The metal cluster will be modeled as an infinitely deep spherical potential well with the represented by an infinitely high spherical barrier. Let us place this barrier in the center of the spherical cluster to simplify the calculations. The simple Schrodinger equation, containing only the interaction of the electrons with the static potential and the kinetic energy term and neglecting any electron-electron interaction, can then be solved analytically, the solutions for the radial wave functions being linear combinations of spherical Bessel and Neumann functions. 

The bound-state energies and eigenfunctions can be obtained by solving the Schrodinger equation with boundary conditions that the radial wave function vanishes at both ends... 

For H2, accurate theoretical calculations3 of the vibration-rotation energy levels have been done by solving the radial differential equation (4.11) using numerical integration. The potential-energy function used is that found from a 100-term variational electronic wave function. 

All of the information that was used in the argument to derive the >2/1 arrangement of nuclei in ethylene is contained in the molecular wave function and could have been identified directly had it been possible to solve the molecular wave equation. It may therefore be correct to argue [161, 163] that the ab initio methods of quantum chemistry can never produce molecular conformation, but not that the concept of molecular shape lies outside the realm of quantum theory. The crucial structure-generating information carried by orbital angular momentum must however, be taken into account. Any quantitative scheme that incorporates, not only the molecular Hamiltonian, but also the complex phase of the wave function, must produce a framework for the definition of three-dimensional molecular shape. The basis sets of ab initio theory, invariably constructed as products of radial wave functions and real spherical harmonics [194], take account of orbital shape, but not of angular momentum. 

The external form of the radial wave function is a solution of (4.57) with the potential —z/r. The uncharged target (such as an atom) is a special case z = 0. It is convenient to rewrite the radial equation in terms of the variable... 

The radial integral equations (4.121) are solved for each partial wave L and the half-off-shell solutions substituted in the equivalent of (4.118) for the T matrix. The on-shell solutions are in fact the Tl of (4.115), from which the scattering amplitude and cross sections can be calculated. 

However, the numerical treatment of such explicitly time-dependent basis states would be time consuming compared to the treatment of bound states. Thus we search for a further simplification of by investigating the asymptotic behavior of Coulomb wave functions [42]. For rAe tt the radial wave function z/j. / is nearly independent of e and may be considered constant for integration. For et tt the exponential function in equation (18) is nearly independent of e. In both cases in equation (18) may be replaced by... 

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